极罕遇地震作用下无预紧力变摩擦惯容器消能体系抗震性能分析

Seismic Performance Analysis of Energy Dissipation Building with Non-preload Variable Friction Inerter Under Extremely Rare Earthquake

摘要

针对传统摩擦阻尼器在极罕遇或近场脉冲型地震作用下需施加较大预紧力以提供足够的阻尼力，且可能引发耐久性问题这一现状，提出一种无预紧力变摩擦惯容器（NVFI）。首先，对NVFI的组成、工作原理及力学模型进行介绍，指出其通过滚珠丝杠驱动弹簧施加可变正压力，从而在摩擦片之间形成随运动变化的变摩擦阻尼力，呈现出“蝴蝶形”滞回曲线特征。其次，建立NVFI的恢复力计算公式，并以单自由度结构为算例，开展各级地震作用下的非线性动力时程分析，验证NVFI的减震效果和能量耗散能力。研究结果表明：NVFI可在无预紧力条件下实现大吨位的变摩擦阻尼力，有效降低结构位移、速度及加速度响应，且对近场脉冲型地震作用下的减震效果尤为突出。在设防地震水平下，NVFI可分别降低远场、近场脉冲型及非脉冲型地震作用下结构位移响应的46%、56%和34%，同时表现出较好的速度与加速度减震性能。能量分析显示，NVFI可显著降低结构输入的地震能量，改变结构基本频率，使之避开外界地震主频的影响范围。此外，通过分析质量比在0～0.5变化对结构动力响应的影响可知，质量比主要影响结构的速度和加速度响应，对位移响应影响相对较小；NVFI的减震与耗能性能主要依赖其优异的变摩擦阻尼特性，而惯容机制主要是降低地震输入能。

Abstract

Objective Traditional energy dissipation technologies provide an effective solution to mitigate the seismic responses of buildings. There are two types of energy dissipation devices based on their operational characteristics: velocity-dependent dampers and displacement-dependent dampers. Friction dampers (FDs), as a category of displacement-dependent energy dissipation devices, exhibit several common advantages, such as good energy dissipation capacity, satisfactory mechanical performance, and ease of fabrication and installation. Therefore, they have received extensive attention from researchers in recent years. Buildings can require large damping forces under extremely rare or near-fault earthquake events, which necessitate that existing FDs apply a high preload to deliver sufficient reaction forces. However, introducing an excessive preload force can be impractical and uneconomical for current FDs. For example, the damping force of existing FDs can need to reach 1 000 kN for buildings subjected to a severe or near-fault earthquake event, resulting in a required preload force of 10 000 kN when the friction coefficient is assumed to be 0.1. In addition, FDs with a specified preload force still face durability issues such as cold bonding, cold solidification, and preload relaxation. Therefore, this study aims to develop a non-preload variable friction inerter (NVFI), which provides satisfactory damping force and significant energy dissipation without relying on preload force. Methods The proposed NVFI mainly consisted of a ball screw, rotational plate, friction plate, spring, and two thrust bearings. One terminal of the ball screw was fixed to the structure using an ear plate. The ball screw of the NVFI generated axial motion when the structure reciprocally shook under seismic earthquakes, and the springs were driven to reciprocal motion, resulting in a variable positive pressure of the friction plate. Therefore, the butterfly-shaped hysteretic behavior of the proposed NVFI was found based on the friction mechanism mentioned above. Then, the restoring force formula of the proposed NVFI was further established. Then, seismic performance mitigation of a single-degree-of-freedom (SDOF) system under different hazard levels was conducted to evaluate the effectiveness and advantages of the proposed NVFI quantitatively. A 5% damping SDOF system with a mass of 50 660 kg and elastic stiffness of 2 000 kN/m was adopted as the analytical model. A Bouc‒Wen elastoplastic model was employed in the SDOF system with a yield strength of 24 kN and post-elastic stiffness of 200 kN/m. The SDOF system with and without the proposed NVFI was considered and denoted as SDOF‒NVFI and SDOF, respectively. Three groups of different ground motion records, including far-field, near-fault pulse, and near-fault non-pulse ground motion records, were selected to perform the nonlinear dynamic analysis, and the peak ground acceleration (P_GA) scaled to multiple intensity levels was 0.2g, 0.4g, and 0.6g for design level earthquake (DLE), maximum considered earthquake (MCE), and extremely rare earthquake (ERE), respectively. Results and Discussions The results illustrated that NVFI significantly reduced the displacement, velocity, and acceleration responses of the SDOF systems subjected to different earthquake records at different hazard levels. The average displacement reduction ratios were 46%, 56%, and 34% for the SDOF‒NVFI subjected to far-field, near-fault pulse, and near-fault non-pulse ground motions at the DLE hazard level, respectively. Similar reductions were also observed in the velocity and acceleration results. Compared to the results of the far-field and near-fault non-pulse ground motions, the displacement and velocity responses of the SDOF systems subjected to near-fault pulse ground motions were more effectively decreased using the proposed NVFI while maintaining a basically approximate acceleration mitigation effect. This is attributed to the fact that the proposed NVFI exhibits good energy dissipation capacity, which was induced by its unique friction mechanism. The satisfactory and variable friction force made the NVFI more suitable for buildings under seismic earthquakes with strong uncertainty, especially for near-fault pulse-such as earthquake events. On the other hand, the seismic input energy of the SDOF system was also decreased through the proposed NVFI. The seismic input energy of the SDOF‒NVFI system was less than 15 kJ at the MCE hazard level, while the seismic input energy of the SDOF system was greater than 24 kJ. This indicated that the fundamental frequency of the SDOF system can be effectively shifted from the dominant frequency of external disturbance by introducing the proposed NVFI, thus improving the overall performance of the structure. Finally, the parameter analysis of the SDOF‒NVFI systems considering different inertance-to-mass ratios was further performed to reveal the influence of the inerter mechanism from the proposed NVFI. The results illustrated that increasing the inertance-to-mass ratio of the proposed NVFI has a significant influence on the velocity and acceleration responses of the SDOF systems at the DLE hazard level while showing a slight but visible influence on the displacement response. Specifically, more than a 10% increase in the velocity and acceleration responses of the SDOF systems subjected to far-field ground motions at the DLE hazard level was observed, with the inertance-to-mass ratio increasing from 0.1 to 0.5. Conclusions The results indicated that the energy dissipation of the SDOF system primarily depends on the superior energy dissipation capacity of the NVFI due to the effectiveness of its variable friction mechanism, whereas the inerter serves only to transform the seismic input energy.

Graphical abstract

关键词

无预紧力 / 变摩擦 / 惯容器 / 极罕遇地震 / 抗震性能

Key words

non-preload / variable friction / inerter / extremely rare earthquake / seismic performance

本刊网刊

近年来，汶川、土耳发生地震的震中烈度都远超过其设防烈度，造成极其惨烈的地震灾害。因此，在超烈度极罕遇地震作用下，如何减轻灾害、防止结构倒塌破坏，成为国内外关注的热点^[1]。第5代《中国地震动参数区划图》为极罕遇地震作用的分析提供计算依据^[2‒3]。

极罕遇地震作用下，结构的水平位移可能会显著增加，加剧结构损坏及与相邻结构碰撞的可能性^[4]；研究表明，结构最大层间位移角超过规范要求时，无法满足正常使用功能阶段的要求^[5‒6]。虞终军等^[7]发现，罕遇、极罕遇地震作用会引起较大的结构残余层间位移角及损伤。吕大刚^[8]、王从^[9‒10]、徐铭阳^[11]和Mao^[12]等发现巨震下的倒塌概率为54.7%，说明巨震下结构倒塌概率比常规地震更大。而采取合适的减隔震控制措施将成为应对极罕遇地震作用的有效手段，包括采用惯容与黏滞阻尼装置降低结构响应^[13‒15]，以及利用多级屈服与损伤控制阻尼器提升结构韧性^[16‒18]。

建筑消能减震技术规程^[19]要求阻尼器在小震、中震和大震都具备良好的变形能力和消耗地震能量的能力，其极限位移应大于消能器设计位移的120%，速度相关型阻尼器极限速度应大于设计速度的120%。在极罕遇地震作用下，阻尼器需满足更高的性能指标，提供更大的阻尼力、设计位移的冗余度，才能确保其在极罕遇地震作用下性能不失效或不发生破坏。因此，越来越多的学者在此方面开展深入研究。秦凯等^[20]提出一种可使耗能能力放大的黏滞消能伸臂装置，在罕遇地震作用下能有效地控制主体结构的层间位移角、基底剪力等地震响应和塑性损伤，综合提高阻尼伸臂系统的能量耗散效率；Huang等^[4]研究表明，在极罕遇地震作用下，质量调谐阻尼器需串联或并联基础隔震，才会有较好的减震效果；Miyamoto等^[21]考虑极罕遇地震，进行全面地分析和实验研究，评估黏滞阻尼器结构在极罕遇地震下的倒塌性能；Wu等^[22]考虑强震，对建筑-阻尼器系统的抗震性能及阻尼器失效的影响进行了系统评价，包括屈曲约束支撑（BRB）、摩擦阻尼器（FD）、黏滞阻尼器（VD）和黏弹性阻尼器（VED），结果表明，在强震作用下，建筑-阻尼器系统响应较大，存在阻尼器失效的可能。

传统摩擦阻尼器通过预紧力螺栓施加预紧力^[23‒25]提供滑动摩擦力，若起滑力较大，阻尼器在小、中震中可能无法发挥耗能作用；若起滑力较小，则在大震、极罕遇地震中无法提供充分的耗能能力，进而难以取得预期的减震效果，且阻尼器位移容易超过其极限能力^[26‒27]。此外，长期处于较大预紧力作用下的摩擦材料之间，易出现冷黏结、冷凝固和预紧力易松弛等问题，导致其力学性能难以准确预测^[28]。陈家川等^[29]提出一种无预紧力变摩擦惯容器，给出其力学简化模型，并通过拟静力试验验证其作用机理和力学模型的正确性；推导了简谐荷载作用下NVFI单自由度减震体系的位移解析解和动力放大系数，并进行相关参数分析。研究表明：该消能装置无须施加预紧力，可随位移变化提供变摩擦力，可自适应调节阻尼力大小，克服了传统摩擦阻尼器存在的问题，具备较强的耗能能力。为促进其在实际工程中应用，有必要进一步探究NVFI减震体系的减震控制效果。

本文在简单介绍NVFI工作机理和力学模型的基础上，对NVFI减震体系的减震控制效果、滞回特性及累积耗能进行分析，同时探究质量比变化对控制效果的影响，进而揭示NVFI应对极罕遇近断层脉冲型地震作用的有效性，为其工程应用提供依据。

1 NVFI的力学模型

NVFI是利用弹簧的压缩变形在摩擦片与圆环形钢板之间提供正压力来产生摩擦力。地震作用下，滚珠丝杠驱动弹簧，其对摩擦片施加的正压力不断变化，从而使摩擦力也相应变化。同时，基于滚珠丝杠机构的位移和阻尼力放大效应来消耗地震能量，NVFI示意图如图1所示。

根据陈家川等^[29]研究，NVFI提供的阻尼力为：

F_{\mathrm{N}\mathrm{V}\mathrm{F}\mathrm{I}}=F_{\mathrm{d}}\mathrm{s}\mathrm{g}\mathrm{n}(\dot{x})+m_{\mathrm{d}}\ddot{x}+k_{\mathrm{d}}x（1）

F_{\mathrm{d}}=\mu {\alpha }_{\mathrm{n}}{k}_{\mathrm{d}}\left|x\right|=\frac{\mathrm{4}}{\mathrm{3}}\mathrm{\pi }\mu \frac{\mathrm{1}}{\eta L_{\mathrm{d}}}·\frac{r_{\mathrm{1}}^{\mathrm{2}}+r_{\mathrm{1}}^{}{r}_{\mathrm{2}}^{}+r_{\mathrm{2}}^{\mathrm{2}}}{r_{\mathrm{1}}^{}+r_{\mathrm{2}}^{}}{k}_{\mathrm{d}}\left|x\right|（2）

m_{\mathrm{d}}=\frac{\mathrm{2}{\mathrm{\pi }}^{\mathrm{2}}(r_{\mathrm{1}}^{\mathrm{2}}+r_{\mathrm{2}}^{\mathrm{2}})}{\eta L_{\mathrm{d}}^{\mathrm{2}}}{m}_{\mathrm{0}}（3）

式（1）～（3）中：F_NVFI为NVFI阻尼力；F_d为随位移变化的放大摩擦力；m_d为放大后质量；F_d、m_d均与构造参数有关；x、\dot{x}和\ddot{x}分别为位移、速度和水平加速度；\mu为摩擦系数；r{}_{\mathrm{1}}和r{}_{\mathrm{2}}分别为圆环形钢板的内半径和外半径；\eta为滚珠丝杠传动效率；k_d为弹簧刚度；m₀为圆环形钢板质量；L_{\mathrm{d}}为滚珠丝杠的导程；\mathrm{s}\mathrm{g}\mathrm{n}(\dot{x})为关于滚珠丝杠速度的符号函数，当\dot{x}\le \mathrm{0}时，\mathrm{s}\mathrm{g}\mathrm{n}(\dot{x})=-\mathrm{1}，当\dot{x}>\mathrm{0}时，\mathrm{s}\mathrm{g}\mathrm{n}(\dot{x})=\mathrm{1}，当\dot{x}=\mathrm{0}时，丝杠处于静止状态。

NVFI简化力学模型如图2所示。NVFI的滞回模型如图3所示，由具有“蝴蝶形”滞回曲线的放大摩擦力F_{\mathrm{d}}\mathrm{s}\mathrm{g}\mathrm{n}(\dot{x})、正刚度效应的弹簧弹性力k_{\mathrm{d}}x及负刚度效应的放大惯性力m_{\mathrm{d}}\ddot{x}这3部分叠加而成。

2 NVFI单自由度减震体系的计算模型及地震波选取

2.1 计算模型

陈家川等^[29]在研究中，通过拟静力试验验证了所提出的NVFI力学模型。试验结果表明，该模型能够较好地描述NVFI装置在不同加载条件下的力-位移关系和耗能特性，为本文的理论分析奠定了基础^[29]。为进一步验证NVFI力学模型在动力分析中的适用性，并提高计算效率，本文选取单自由度结构（SDOF）作为分析对象。NVFI单自由度减震体系（SDOF-NVFI）计算模型如图4所示，其在地震作用下的运动方程为：

\left\{\begin{array}{l}m\ddot{x}+c\dot{x}+kx+F_{\mathrm{N}\mathrm{V}\mathrm{F}\mathrm{I}}=-m{\ddot{x}}_{\mathrm{g}},\\ F_{\mathrm{N}\mathrm{V}\mathrm{F}\mathrm{I}}=F_{\mathrm{d}}\mathrm{s}\mathrm{g}\mathrm{n}(\dot{x})+m_{\mathrm{d}}\ddot{x}+k_{\mathrm{d}}x\end{array}\right.（4）

式中，m为单自由度体系的质量，c为阻尼系数，k为单自由度体系的刚度，{\ddot{x}}_{\mathrm{g}}为地震输入加速度。

2.2 地震波选取

为研究在不同场地地震波作用下NVFI减震耗能的有效性，选取FEMA P695中推荐的近断层脉冲、近断层非脉冲及远场地震动3组（每组各7条），地震记录参数如表1所示。表1中，P_GA为地面加速度峰值，P_GV为地面速度峰值。分析过程中对地震波进行归一化处理，具体分析时可采用调幅的方法调整设防地震、罕遇地震和极罕遇地震对应的加速度峰值，各组地震波平均加速度反应谱如图5所示。由图5可知：在0～0.5 s内，3种类型地震动平均加速度反应谱值相近，在0.5～4.0 s内的大部分周期对应的平均加速度反应谱值大小关系为：近断层脉冲型地震大于远场地震动，远场地震动大于近断层非脉冲型地震动。

3 NVFI单自由度减震体系的减震效果分析

取单层框架结构的质量、刚度和阻尼比分别为50 660 kg、2 000 kN/m和5%^[30]，结构非线性模型采用Bouc-Wen模型，屈服后刚度比0.1，屈服位移12 mm。NVFI摩擦系数0.2，圆环形钢板内径和外径分别为0.1和0.2 m，圆环形刚板质量5 kg，丝杠导程0.02 m，弹簧刚度为15 kN/m。

3.1 减震控制效果分析

通过计算分析可知，3组地震波在设防地震（0.2g）、罕遇地震（0.4g）和极罕遇地震（0.6g）作用下，NVFI减震体系的最大位移、速度及加速度的结构最大反应平均减震率如图6所示。

由图6可知：1）在相同地震动加速度峰值输入时，对于3组地震波整体而言，最大位移平均减震效果优于最大速度和最大加速度平均减震效果；2）脉冲型和远场地震波的减震效果要明显优于近断层非脉冲型地震，因为单自由度结构模型的周期为1 s，此处所选脉冲型地震动和远场地震动的平均反应谱值要大于非脉冲地震波，同时，脉冲型地震动有脉冲效应，导致脉冲型地震下的结构位移大于非脉冲下的位移，由于NVFI具有变摩擦特性，位移越大，NVFI出力越大，对脉冲型和远场地震波控制效果比非脉冲型好；3）随着输入地震加速度峰值的增大，近断层脉冲型地震动的最大位移、速度和加速度平均减震率均呈增大趋势，极罕遇地震下减震效果尤为明显；而对于近断层非脉冲地震动和远场地震动，最大位移平均减震率呈增大趋势，最大速度平均减震率变化不明显，最大加速度减震率呈下降趋势，表明NVFI对极罕遇脉冲型地震取得了优异的减震控制效果。

3.2 地震反应时程分析

在极罕遇脉冲型Erzican、近断层非脉冲Imperial Valley-06、远场Kocaeli地震波作用下（0.6g），有控和无控结构的位移、速度和加速度的时程曲线如图7所示。

由图7可见：在3组极罕遇地震下，NVFI对位移、速度和加速度的整体减震效果均较好，位移、速度整体控制效果优于加速度；近断层脉冲型地震动和远场地震动的整体减震效果优于非脉冲型地震动，且随着输入地震加速度峰值的增加，位移、速度和加速度整体控制效果有所提高。

3.3 NVFI滞回特性分析

3组地震波作用下NVFI滞回曲线如图8所示。由图8可知，NVFI呈现出明显的位移相关性变摩擦效应，滞回曲线与第3.1节理论力学模型吻合。

3.4 耗能能力分析

无控结构和NVFI减震体系累积耗能时程曲线分别如图9和10所示。由图9和10可知：无控结构中，地震波输入能量主要依赖结构变形和阻尼耗能；有控结构中，地震总输入能量相对于无控结构会减小。这是由于NVFI放大质量改变了结构自振周期。同时，地震输入能主要由NVFI耗散，结构变形能与无控结构相比大幅减小，结构层间位移也大幅降低。在极罕遇地震下，地震总输入能与NVFI耗能曲线之间的距离最小，表明NVFI取得了最佳的耗能减震效果。

3组地震波在不同加速度峰值作用下，NVFI平均耗能率（NVFI耗能与总输入能之比）如图11所示。由图11可知：对于3组不同类型地震波，NVFI均取得了50%以上的耗能效果。同一类地震波，随着输入加速度峰值增大，NVFI平均耗能率呈现上升的趋势，对于极罕遇脉冲型地震动更为明显。说明NVFI能较好地耗散极罕遇及近断层脉冲型地震动的输入能量，体现了稳定可靠的减震性能。

4 NVFI与惯容器减震效果对比分析

当NVFI的弹簧刚度为0时，则NVFI的阻尼力等于其放大惯性力，此时NVFI相当于一个惯容器（Inerter）。对无控结构、Inerter减震体系和NVFI减震体系反应进行对比分析，探索Inerter质量变化对最大减震率、整体减震效果的影响，并与NVFI减震效果进行对比。设NVFI的弹簧刚度为100 kN/m，取质量比（β=m_d/m）变化区间为0～0.5。

4.1 减震效果对比

近断层和远场地震作用下质量比变化对减震效果的影响规律分别如图12和13所示。由图12和13可知：在不同地震类型作用下，两种减震体系中，随质量比增大，位移、速度和加速度峰值减震率大都明显增大；相比Inerter，NVFI的减震控制效果更为明显，且随质量比增大NVFI的位移减震效果增加不明显，表明其主要依靠变摩擦阻尼耗能。

4.2 累积耗能对比

远场Kocaeli波作用下质量比分别为0.1、0.5时，Inerter减震体系和NVFI减震体系的累积耗能时程曲线分别如图14和15所示。

由图14和15可知：惯容器基本不耗散地震输入能量，但随质量比增大，惯容器和NVFI的放大质量在一定程度上延长了结构自振周期，使减震体系的总输入能有所减小；Inerter减震体系中，地震输入能主要由结构变形耗散；NVFI减震体系中，地震输入能主要由NVFI和结构变形耗散，进一步验证了NVFI中变摩擦机制的必要性及耗能有效性。

5 结论

本文在前期对简谐荷载作用下NVFI单自由度减震体系的位移解析解和动力放大系数的研究基础上，简介了NVFI的工作机理和力学模型，对NVFI单自由度减震体系进行了减震控制效果分析，并研究了质量比变化对减震效果的影响。主要结论如下。

1）地震作用下，NVFI无须施加预紧力就可提供变阻尼力，具有摩擦力、位移和惯性质量放大效应，呈现出“蝴蝶形”滞回特征，克服了传统摩擦阻尼器的不足，可自适应极罕遇地震和近断层脉冲型地震对大吨位阻尼器的减震需求。

2）在近断层脉冲、非脉冲及远场地震作用下，对设防、罕遇、极罕遇地震，NVFI减震体系的位移、速度、加速度均可取得较好的减震控制效果，体现了稳定可靠的减震性能，尤其对极罕遇脉冲型地震可取得更佳的减震控制效果，具有较好的工程应用前景。

3）质量比变化区间为0～0.5时，单纯惯容器基本不耗散地震输入能量，但随质量比增大，惯容器和NVFI的放大质量在一定程度上延长了结构自振周期，使减震体系的总输入能有所减小；NVFI减震体系中，地震输入能主要由NVFI和结构变形耗散，进一步验证了NVFI中变摩擦机制的必要性及耗能有效性。

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基金资助

国家重点研发计划项目(2022YFC3801201)

国家重点研发计划项目(2019YFE0112500)

国家自然科学基金项目(52378499)

国家自然科学基金项目(52308488)

国家自然科学基金项目(52078150)

广东省基金面上项目(2023A1515010072)